Does this property characterize a space as Hausdorff?

Question 1

As a result of this question, I’ve been thinking about the following condition on a topological space $Y$ :

For every topological space $X$ , $E\subseteq X$ , and continuous maps $f,g\colon X\to Y$ , if $E$ is dense in $X$ , and $f$ and $g$ agree on $E$ (that is, $f(e)=g(e)$ for all $e\in E$ ), then $f=g$ .

If $Y$ is Hausdorff, then $Y$ satisfies this condition. The question is whether the converse holds: if $Y$ satisfies the above condition, will it necessarily be Hausdorff?

If $Y$ is not at least $T_1$ , then $Y$ does not have the property: if $u,v\in Y$ are such that $u\neq v$ and every open neighborhood of $u$ contains $v$ , then let $X$ be the Sierpinski space, $X=\{a,b\}$ , $a\neq b$ , with topology $\tau=\{\emptyset,\{b\},X\}$ , $E=\{b\}$ , let $f,g\colon X\to Y$ be given by $f(a)=f(b)=v$ , and $g(a)=u$ , $g(b)=v$ . Then both $f$ and $g$ are continuous, agree on the dense subset $E$ , but are distinct.

My attempt at a proof of the converse assumes the Axiom of Choice and proceeded as follows: assume $Y$ is $T_1$ but not $T_2$ ; let $u$ and $v$ be witnesses to the fact that $Y$ is not $T_2$ , let $\mathcal{U}\_s$ and $\mathcal{V}\_t$ be the collection of all open nbds of $s$ that do not contain $t$ , and all open nbds of $t$ that do not contain $s$ , respectively. Construct a net with index set $\mathcal{U}\_s\times\mathcal{V}\_t$ (ordered by $(U,V)\leq (U',V')$ if and only if $U'\subseteq U$ and $V'\subseteq V$ ) by letting $y_{(U,V)}$ be a point in $U\cap V$ (this is where AC comes in). Let $E=\{y_{(U,V)}\mid (U,V)\in\mathcal{U}\_s\times\mathcal{V}\_t\}$ , and let $X=E\cup\{s\}$ . Give $X$ the induced topology; let $f\colon X\to Y$ be the inclusion map, and let $g\colon X\to Y$ be the map that maps $E$ to itself identically, but maps $s$ to $t$ .

The only problem is I cannot quite prove that $g$ is continuous; the difficulty arises if I take an open set $\mathcal{O}\in \mathcal{V}_t$ ; the inverse image under $g$ is equal to $((\mathcal{O}\cap X)-\{t\})\cup\{s\}$ , and I have not been able to show that this is open in $X$ .

So:

Does the condition above characterize Hausdorff spaces?

If not, I would appreciate a counterexample. If it does characterize Hausdorff, then ideally I would like a way to finish off my proof, but if the proof is unsalvageable (or nobody else can figure out how to finish it off either) then any proof will do.

Added: A little digging turned up this question raised in the Problem Section of the American Mathematical Monthly back in 1964 by Alan Weinstein. The solution by Sim Lasher gives a one paragraph proof that does not require one to consider $T_1$ and non- $T_1$ spaces separately.

Question 2

[This answer moved from this question on Arturo’s advice]

I think the following goes a long way towards proving a converse:

Let $(Y, T)$ be any T1 topological space with at least two points and let $a$ and $b$ be distinct points in $Y$ .

Let $X = Y\setminus\{b\}$ .
Let $f: X \to Y$ be the inclusion of $X$ in $Y$ .
Let $g: X \to Y$ agree with $f$ on $X\setminus\{a\}$ and $g(a) = b$ .

Finally, define the topology on $X$ to be the coarsest topology that makes both $f$ and $g$ continuous.

With these assumptions it turns out that $X\setminus\{a\}$ is dense in $X$ if and only if $a$ and $b$ do not have disjoint neighbourhoods in $Y$ .

To see that this is true, let us construct a base of the topology on $X$ .
To make $f$ continuous we only need to take the subspace topology. Since
X is open in Y this is $S_1 = \{ G \in T \mid b \notin G \}$ . To also make $g$ continuous we need to add the open neighbourhoods of $b$ , with $b$ replaced by $a$ . This gives $S_2 = \{ (H\setminus\{b\} \cup \{a\} \mid H \in T, b \in H \}$ .
Now $S_1 \cup S_2$ is a subbase of the topology on $X$ .
Since $S_1$ and $S_2$ are already closed under finite intersection, and each covers $X$ , we can say that
$B = \{ G \cap H \mid G \in S_1, H \in S_2 \}$ is a base of the topology.

Then (remembering that finite sets are closed in Y) we find that the following are all equivalent:

$X\setminus\{a\}$ is not dense in $X$
$\{a\}$ is open in $X$
$\{a\} \in B$
there are $G \in S_1, H \in S_2$ such that $G \cap H = \{a\}$
there are $G \in S_1, H \in S_2$ such that $a \in G$ and $G \cap (H\setminus\{a\}\cup\{b\}) = \oslash$
$a$ and $b$ have disjoint neighbourhoods in $(Y, T)$ .

Does this property characterize a space as Hausdorff?

Answer

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